We have been pulling our hair out for several days due to this issue. Office 365 Exchange admin center gives the following error whereas the new admin center worked well.
when you click that “Exchange” blade from the Office 365 admin center, it usually takes you to the classic Admin center which we still need for some functions that new Admin center doesn’t have.
After lots of struggle, we managed to figure out the Root cause and reported to Microsoft through an incident.
Root cause: Group based access assignments in Privileged Identity Management.
Workaround: We had assigned Azure AD Roles such as Global Administrator, Exchange Administrator via Group based PIM which did not work properly with classic EAC. Assigning Direct permissions fixed this and we managed to open the classic console immediately, right after the direct assignment. If you are facing the same, try to get rid of “Group Assignments” for Exchange Admins at least for the time being and go for “Direct Assignments”
Official reference: https://docs.microsoft.com/en-us/azure/active-directory/roles/groups-concept
I will update this post up-on Microsoft’ support responses.
Microsoft Cloud Infrastructure Technical Journal 
Hi,
I see that you ran into this a year and a half ago. Did you ever hear back from Microsoft? I ran into the same issue and have been advising our team to use new admin center. However, we’re also seeing this issue in the new admin center specifically attempting to manage transport rules.
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